Astronomy&Astrophysicsmanuscriptno.3555pol (cid:13)c ESO2008 February5,2008 Modeling optical and UV polarization of AGNs I. Imprints of individual scattering regions Rene´ W.Goosmann1,2 andC.MartinGaskell3 1 AstronomicalInstituteoftheAcademyofSciences,BocˇniII1401,14131Prague,CzechRepublic e-mail:[email protected] 2 ObservatoiredeParis-Meudon,5placeJulesJanssen,92190Meudon,France 7 3 DepartmentofPhysics&Astronomy,UniversityofNebraska,Lincoln,NE68588-0111,USA 0 e-mail:[email protected] 0 2 ReceivedJune2005;acceptedDecember2006 n a ABSTRACT J Context.SpectropolarimetryofAGNsisapowerfultoolforstudyingthestructureandkinematicsoftheinnerregionsofquasars. 6 Aims.WewishtoinvestigatetheeffectsofvariousAGNscatteringregiongeometriesonthepolarizedflux. Methods.Weintroduceanew,publiclyavailableMonteCarloradiativetransfercode,S,whichmodelspolarizationinducedby 3 scatteringofffreeelectronsanddustgrains.WemodelavarietyofregionsinAGNs. v Results.Wefindthat theshape of thefunnel of thedustytorushasasignificant impact onthepolarizationefficiency. Acompact 2 toruswithasteepinnersurfacescattersmorelighttowardtype-2viewinganglesthanalargetorusofthesamehalf-openingangle, 7 θ .For θ < 53◦,thescattered light ispolarized perpendicularly tothesymmetryaxis, whilst for θ > 60◦ itispolarizedparallel 0 0 0 0 to the symmetry axis. In between these intervals the orientation of the polarization depends on the viewing angle. The degree of 7 polarizationrangesbetween0%and20%andiswavelengthindependentforalargerangeofθ .Observedwavelength-independent 0 0 opticalandnear-UVpolarizationthusdoesnotnecessarilyimplyelectronscattering.Spectropolarimetryatrest-framewavelengths 5 lessthan2500 Åmaydistinguishbetweendust andelectronscatteringbutisnotconclusiveinallcases.Forpolar dust,scattering 0 spectraarereddenedfortype-1viewingangles,andmadebluerfortype-2viewingangles.Polarelectron-scatteringconesarevery / h efficientpolarizersattype-2viewingangles,whilstthepolarizedfluxofthetorusisweak. p Conclusions.WepredictthatthenetpolarizationofSeyfert-2galaxiesdecreaseswithluminosity,andconcludethatthedegreeof - polarization should be correlated with the relative strength of the thermal IR flux. We find that a flattened, equatorial, electron- o scattering disk, of relatively low optical depth, reproduces type-1 polarization. This is insensitive to the exact geometry, but the r observedpolarizationrequiresalimitedrangeofopticaldepth. t s a Keywords.Galaxies:active–Polarization–Radiativetransfer–Scattering–Dust : v i X 1. Introduction entation effects (Lawrence & Elvis 1982; De Zotti & Gaskell r 1985). Since then, the dusty-torusmodel has become the stan- a One of the foremost problems in AGN research is that the in- dard unified model (see Antonucci 1993) dividing AGNs into nermostregionsofAGNscannotberesolvedintheopticaland twosub-types:“type-1”AGNswhichareseenclosetoface-on, UVwithcurrenttechnology.However,thelightofAGNsispo- and“type-2”AGNswhichare seen close toedge-on.Intype-1 larizedoverabroadwavelengthrange,andthisallowsustoput AGNs the central energysource and its surroundings(e.g., the importantconstraintsonthegeometryoftheemittingandscat- BLR) can be seen, whilst in type-2 AGNs the torus blocksour tering regions. Spectropolarimetric observations giving the de- direct view of these inner regions. While this obscuration and tailedwavelengthdependenceofthepolarizedfluxgivefurther the IR emission from the torus are the most obviouseffects of cluestothenatureofthepolarizingmechanism. the torus, scattering from the dust will add polarized flux. The OurinferencesoftheinnermoststructuresofAGNshaveso polarizationspectrumofanopticallythickdustytorushasbeen farbeenobtainedindirectly.Rowan-Robinson(1977)suggested the subject of several modeling projects (Kartje 1995, Wolf & thatAGNsaresurroundedbyadustytorusandinthesamepaper Henning1999,Watanabeetal.2003). he gives a suggestion by M. V. Penston that Seyfert2 galaxies areseenclosetoedge-onso thattheactivenucleusisobscured WhenDibai&Shakhovskoy(1966)andWalker(1966)dis- by the torus. Support for this picture came from the important covered optical polarization of AGNs, it was initially taken to discoveryby Keel (1980) that Seyfert1 galaxies(active galax- be evidenceofopticalsynchrotronemission,since synchrotron ies showing a broad-line region; BLR) are preferentially seen radiation has a high intrinsic polarization. However, Angel et face-on.Keel (1980) also investigatedabsorptioneffects inside al.(1976)foundtheBalmerlinesinNGC1068tobepolarized thehostgalaxiesandemphasizedtheneedofadditionalnuclear similarlytothecontinuum,thusimplyingthatscatteringwasre- absorption in Seyfert galaxies with respect to normal spirals. sponsible for the polarization of both the lines and continuum. Keel’sworkledtofurtherconfirmationoftheimportanceofori- The difference they found in polarization between the narrow- lineregion(NLR)andBLRplacesthescatteringregionoutside Sendoffprintrequeststo:Rene´W.Goosmann theBLR,butinsidetheNLR 2 Goosmann&Gaskell:ModelingAGNPolarization When light is scattered, the angle of polarization depends wedescribeS,anewgeneral-purpose,publicly-available,1 onthedirectionofthelastscattering,sooneexpectstheangleof MonteCarlocodeformodelingwavelength-dependentpolariza- polarizationtoberelatedtothestructureoftheAGN.Stockman, tioninawidevarietyofscenarios,andwepresentsomeresults Angel,&Miley(1979)madetheseminaldiscoverythatforlow- ofourstudyofAGNpolarization. polarization,highopticalluminosity,radio-loudAGNs,the op- InthispaperweconfineourselvestousingSforcalcu- tical polarization position angles tend to align parallel to the latingthepolarizationimprintsofbasicconstituentsoftheuni- large-scale radio structure. Although they interpreted this as a fiedscheme.Wecomputethepolarizationspectrumofdustytori consequence of optical synchrotron emission, they also sug- with various geometries and opening angles, and we consider gestedthatpolarizationfromanoptically-thin,non-spherically- scattering in polar conesand electron disks. We investigatethe symmetricscatteringregionnearthesourceofopticalradiation effectsofgeometricalshapeandopticaldepthofgivenregions. wasanotherpossibility. Noneofourmodelsareintendedtoreproduceobservationalpo- larimetricdataforanyspecificobject.Rather,wewanttoinves- Antonucci(1982)pointedoutthatwhilstmanyradiogalax- tigategeneralconstraintsonthescatteringregionsandthegeom- ies showed a similar parallel alignmentof the polarizationand etryofAGNs.Wediscussconsequencesfortheobservedpolar- radioaxes,therewas,unexpectedly,apopulationshowingaper- izationdichotomybetweentype-1andtype-2objects.Weleave pendicularrelationship.Itwas subsequentlyshown(Antonucci asidethequestionofinteractionsbetweendifferenttypesofscat- 1983)thatrelatively-radio-quietSeyfertgalaxiesshowasimilar teringregionsforpaperII(Goosmann&Gaskell,inpreparation) dichotomybetweenthepredominantly,butnotexclusively,par- wherewealsoconductmoredetailedmodelingofAGNsinthe allel polarizationin face-ontype-1 Seyfertsand the perpendic- unifiedscheme. ular polarizationof type-2 Seyferts(see Antonucci1993, 2002 The present paper is organized as follows: in section 2 we forreviews).Thesediscoveriesmadeasynchrotronoriginofthe summarize modeling of optical and UV polarization of AGN polarizationmuchlesslikely. and the main results obtained previously. Section 3 describes Polarization perpendicular to the axis of symmetry is eas- our code S. In section 4 we present modeling results for ily producedby scattering off material close to the axis. There equatorial,toroidaldust distributions.Section 5 is dedicated to is good observational evidence for the existence of ionization electronanddustscattering in polardouble-cones.In section 6 cones along the polar axis in numerousobjects (see Kinney et we investigate the polarization signature of equatorial regions al.1991andreferencetherein).Polarscatteringhasbeenpartic- forelectronscattering.Ourresultsarediscussedinsection7and ularlywellstudiedintheSeyfert-1galaxyNGC1068.Antonucci wegivesomeconclusionsinsection8. &Miller(1985)madethekeydiscoverythatthepolarized-flux spectrum can offer a periscope view of type-2 AGNs because much of the polarizedflux originatesinside the torus. Detailed 2. Previouscodesandmodeling HST polarimetry has revealed the polarization structure of the Inthissection,webrieflysummarizesomerecentAGNpolariza- ionizationcones(seeCapettietal.1995a,b;Kishimoto1999). tionmodelingcodeswhichwewillcompareourSmodel- Thedetectionofahiddenbroad-lineregioninNGC1068by ingwith. Antonucci & Miller (1985) was of great importance for AGN Young et al. (1995, 1996), Packham (1997), and Young research since it provided strong support for the unified theo- (2000) developed an analytical radiative transfer model, the ries of AGN activity. More hidden type-1 nuclei have subse- Generic Scattering Model (GSM), for polarization modeling. quently been found by analysis of their polarized-flux spectra ThemodelisbasedontheunifiedAGNmodel.Extendedemis- (seee.g.,Miller&Goodrich,1990;Tran,Miller,&Kay,1992; sionregionscanbedefined,andscatteringprocessesaswellas Hines & Wills, 1993; Kay, 1994; Heisler, Lumsden, & Bailey, dichroicabsorptionareconsidered.Themodeledgeometriesin- 1997; Tran, 2001; Smith et al., 2004). Similar work on the ra- clude toroidal, disk-like, and conical regions of dust and free dio galaxy 3C 321 was done by Young et al. (1996), and Tran electrons.Forscatteringmaterialinmotion,Dopplereffectsare et al. (1999) could identify an active nucleus inside an ultra- included.Themodelisfairlyeffectiveinreproducingspectropo- luminous infra-red galaxy using spectropolarimetry. Recently, larimetric data of Seyfert galaxies (see, for example, Young et hiddentype-1nucleihavealsobeenfoundinfivetype-2quasar al. 1999forMrk509).Inparticular,it reproducesvariationsof candidates(Zakamskaetal.2005). thepolarizationacrossbroademissionlines(Smithetal.2005). Themodelissemi-analyticalandthereforedoesnottakemulti- The new generation of large telescopes is delivering spec- plescatteringintoaccount. tropolarimetryofemissionlineprofileswithgoodvelocityreso- Wolf&Henning(1999)presentaMonte-Carlocodeusedto lution (see, for example, the spectropolarimetry of the Seyfert computethe polarizationobtainedby scattering inside axisym- 1.5 galaxy NGC 4151 presented by Martel 1998, and the at- metric regions. They consider dust and electron scattering for lasofspectropolarimetryofSeyfertgalaxiespresentedbySmith polar double-conesand equatorialtori. In the Monte-Carlo ap- etal.2002). Examinationofvelocity-dependentpolarizationof proach two or more of such componentscan be combined and emissionlinespromisestorevealvaluableinformationaboutthe theresultingpolarizationspectraaremodeledforvariousincli- geometryoftheBLR(Smithetal. 2005).Similarly,spectropo- nations of the system. Aside from spectropolarimetric model- larimetryofquasarabsorptionlineshelpsconstrainthegeometry ing, the code by Wolf & Henning (1999) can also producepo- ofbroadabsorptionlineQSOs(Goodrich&Miller1995,Cohen larizationimages,whichare providedforvarioustorusgeome- etal.1995,Hines&Wills1995,Ogleetal.1999). tries.Animportantelementinthiscodeisthatmultiplescatter- In order to understandthese manyfacets of AGN polariza- ing,which becomesimportantforopticaldepths> 0.1,is con- tion, and their implications for the underlying geometry, theo- sidered accurately by including the dependence of the scatter- reticalmodelingisnecessary.Analyticalapproachestoradiative ingangleandthepolarizationof ascatteredphotononitsinci- transferthathavebeencarriedoutsofararegenerallylimitedto dentStokes vector. For dustscattering, two differentgrain size theconsiderationofsingle-scatteringmodels.Computersimula- tionsareneededtoinvestigatemultiple-scatterings.Inthispaper 1 http://www.stokes-program.info/ Goosmann&Gaskell:ModelingAGNPolarization 3 distributionswereexamined:oneparameterizationrepresenting Galacticdust,andtheotherfavoringlargergrains. Kartje(1995)alsodevelopedaMonteCarloCodeandinves- observer z tigated quasar schemes with either a torusgeometryor conical stratifiedwindsalongthepolaraxis.Inadditiontopolarization last scattering event byscattering,healsoconsiderspolarizationbydichroicextinc- tion due to magnetically-aligneddust grains. For a simple uni- fiedtorusmodelhefindsthatthedominantparameterofthepo- Θ larization, P,isthetorushalf-openingangle:fortype-2objects creation of one can find significant polarization (up to 30%) with a posi- a new photon φ tion angle directed perpendicular to the axis of symmetry; for type-1objectsPisnegligible.Kartjeobtainsanimportantresult y when he investigatesconical stratified-wind regionscontaining free electrons closer to the central source and dust farther out: emission region theamountofpolarizationrangesbetween0%and13%,match- ing observedvalues, and the direction of the E-vectordepends scattering region on the viewing angle in a manner that agrees with the above- scattering events mentioned type-1/type-2 dichotomy. The polarization percent- age can be increased if there is magnetic alignment of dust x grains, but the general dependence of P on the viewing angle Fig.1.Aphotonworkingitswaythroughthemodelspace. seemstobeageometricaleffect. Another Monte-Carlo polarization code is presented by Watanabe et al. (2003). It is applied to modeling of optical region through various scattering processes until they become andnear-infraredspectropolarimetricdataofthetype-2Seyfert absorbed or manage to escape from the model region (Fig. 1). galaxiesMrk463E,Mrk1210,NGC1068,andNGC4388.The The polarization properties of the model photons are given by codecontainselectronanddustscatteringroutinesquitesimilar theirstoredStokesvectors. to thoseusedbyWolf & Henning(1999).Itconsidersmultiple Photonsarecreatedinsidethesourceregions,whichcanbe scatteringanddichroicabsorptionindustytori,spheresaswell realizedbydifferentgeometries.Thecontinuumradiationisnor- aselectronanddustscatteringindouble-conicalregions.Theab- mallysimplydefinedbytheindexαofanF ∝ ν−α powerlaw. ν sorptionandscatteringpropertiesofthedustarecarefullycalcu- TheStokesvectorsofthephotonsareinitiallyset tothevalues latedbyMietheory.Watanabeetal.(2003)examinewavelength- ofcompletelyunpolarizedlight. dependentpolarizationpropertiesfordifferentgeometriesovera Various scattering regions can be arranged around the broad-wavelengthrangeandgiveconstrainsaboutpossiblescat- sources. The programofferse.g. toroidal,cylindrical,spherical teringcomponentswithin theobjectstheyobserved.Theycon- or conical shapes. These regions can be filled with free elec- clude that a combination of dust and electron scattering in po- tronsordustconsistingof“astronomicalsilicate” andgraphite. lar regions can reproducethe optical polarization propertiesof A photon works its way through the model region and gener- Mrk463EandMrk1210.TheslopeofopticalpolarizationNGC allyundergoesseveralscatterings.Theemissiondirections,path 1068isalmostflatfavoringelectronscatteringasthe dominant lengthsbetweenscatteringevents,andthescatteringanglesare polarizing process. For the near-infrared range polarization of computedby MonteCarlo routinesbased onclassical intensity theseobjectscanbemodeledbydichroicabsorptionofaligned distributions. During each scattering event the Stokes vector is dust grains in a torus. However, scattering off Galactic dust in changedby multiplicationwith the correspondingMueller ma- atoruscannotsimultaneouslyreproducethenear-infraredpolar- trix.Fordustscattering,absorptionisimportant,andalargefrac- izationandthetotalflux.Watanabeetal. (2003) hencesuggest tionofthephotonsneverreachesthevirtualobserver.Therele- thatthegrainsizecompositionofAGNsmightbedifferentfrom vantcross sectionsand matrix elementsfor dust scattering and ourGalaxy. absorptionare computedon the basis of Mie theory applied to This list of previous polarization modeling is not exhaus- sizedistributionsofsphericalgraphiteandsilicategrains. tive. For example, Blaes & Agol 1996 and Agol & Blaes If a photon escapes from the model region, it is registered 1996 have presented modeling of the wavelength-dependent by a web of virtual detectors arranged in a spherical geometry polarization signature of accretion disks at the Lyman limit, aroundthesource.Thefluxandpolarizationinformationofeach and Kishimoto (1996) modeled polarization due to electron- detector is obtained by adding up the Stokes parameters of all scattering off clumpy media in polar regions of AGN. Also, a detectedphotons.Ifthemodeliscompletelyaxiallysymmetric new Monte-Carlo model, which includes polarization transfer these can be azimuthally integrated and, if there is plane sym- for the continuum and for broad quasar absorption lines, was metry,thetopandbottomhalvesarecombined.Theobjectcan recently presented by Wang, Wang, & Wang (2006). We have be analyzed in total flux, in polarized flux, percentage of po- restrictedthisbriefreviewtorecentmodelingofpolarizationby larization, and the position angle at each viewing angle. The dustandelectronscattering,sincethisisourprimaryconcernin light travel time of each photon is also recorded, so it is pos- thepresentpaper. sible to model time-dependent polarization (Gaskell, Shoji, & Goosmann,inpreparation). 3. Stokes–anoverview 3.1.MonteCarlomethod,photoninitialization,andsampling ThecomputerprogramSperformssimulationsofradiative thefreepathlength transfer,including the treatmentof polarization,for AGNs and related objects. The code is based on the Monte Carlo method Using the Monte Carlo methodit is possible to generatea ran- andfollowssinglephotonsfromtheircreationinsidethesource domeventxaccordingtoagivenprobabilitydensitydistribution 4 Goosmann&Gaskell:ModelingAGNPolarization p(x).Let p(x)be definedon the interval[0,x ]. We canthen scatteringoffasphericalparticle,thefollowingrelationbetween max constructtheprobabilitydistributionfunctionP(x)andrelateit theincomingandscatteredelectricfieldsholds: toarandomnumber,r,between0and1asfollows: E S (θ) 0 E 1 x k,s = 2 k,i . (6) r= P(x)= p(x′)dx′. (1) E⊥,s ! 0 S1(θ)! E⊥,i! C Z 0 The scattering matrix elements, S (θ) and S (θ), are inde- The constant C is a normalization constant resulting from 1 2 pendentoftheazimuthalangleφ.IncaseofThomsonscattering, integrationoverthewholedefinitioninterval[0,x ].Giventhe max thereabsolutevaluesobeytosimpleanalyticexpressions: randomnumber,thecorrespondingvalueof xforasingleevent isobtainedbyinvertingequation(1).Agooddescriptionofthe MonteCarlomethodcanbefoundinCashwell&Everett(1959). |S (θ)|2 = 1, (7) In the following, we describe the main routines of S and 1 |S (θ)|2 = cos2θ. (8) denote all random numbers computed from equation (1) by r, 2 i withi=1,2,3.... For dust scattering, the albedo and the matrix elements of To generate a model photon, its initial parameters of po- a standard dust grain are calculated from Mie theory (see sec- sition, direction of flight, and wavelength all have to be set. tion 3.3). The albedo at the photon wavelength is compared to Different geometries for the continuum region, broad-line re- gion,andnarrow-lineregionareavailableinS.Assuming a random number,r5, in order to decide whether the photon is absorbedorscattered.Ifthephotonisabsorbeditislost,andthe a constant density of the emitting material, a random position cyclestartsoverwiththegenerationofanewphoton. for the new photonis sampled.The flight directionis givenby Thepolarizationvectorofeachphotonliesperpendicularto two angles, θ and φ, defined with respect to a standard polar itstrajectory,insidetheso-calledpolarizationplane,anddenotes coordinate system. Assuming isotropic emission, the sampling thepreferreddirectionoftheE-vector.Itisdefinedwithrespect equationsfortheanglesareasfollows: toaco-movingcoordinatesystem.Thepolarizationinformation ofeachphotoniscodedbyfourStokesparametersI,Q,U,and θ = arccos(1−2r ), (2) V,representingthe4-dimensionalStokesvector.Weassumethat 1 newlycreatedphotonscomingfromthesourceareunpolarized. φ = 2πr . (3) 2 Hence,theirStokesvectorshavethesimpleform: The wavelength of the photon is sampled according to the I 1 intensityspectrumoverarange[λ ,λ ].Thisleadsto: min max Q 0  = . (9) λ=Heλhrλemαm,iniαn(cid:16)+λλdmmearinxn3(cid:17)o(cid:16)r3tλe,αmsaxth−eλuαmsuina(cid:17)liα1po,wffooerrrααlaw=,i11n.,dex of the intens(i4ty) uannUVgdlWeergiφtoh,eeos00acacchudrosscuabatrlteoeurrinondtgatteihvoeenn:cttu,hrterheefincrtsoflt-imrgohotavttiidnoignre,ccobtoiyorntdhienoafatztehimseyusptthehoma-l spectrum. ton. It rotates the E-vector inside the polarization plane to the Ifweignorescatteringsbackintothebeam,theintensityof position of the new scattering plane (see Fig. 2). Physically, it aphotonbeamtraversingaslabofscatteringmaterialwithpar- doesnotaffectthepolarizationstate,althoughtheStokesvector ticlenumberdensityNandcross-sectionσwilldropbyafactor undergoesthefollowingcoordinatetransformation. of eNσl, with l being the distance traveled inside the scattering region.Fromthis,onecanderivethesamplingfunctionofl: I∗ 1 0 0 0 Iin Q∗ 0 cos2φ sin2φ 0 Qin l= TN1hσelfna(c1to−rr41).is the mean free path length. Depending(o5n) UV∗∗=00 −si0n2φ cos02φ 01UViinn. (10) Nσ The second rotation occurs in the scattering plane by the thescatteringmaterial,theprogramuseseitheradustextinction scatteringangleθ.ThechangeoftheStokesvectorisdetermined cross-sectionσ computedfromMietheoryor,incaseofelec- ext bytheMuellermatrix,whichforscatteringoffasphericalparti- tronscattering,theThomsoncrosssectionσ . ES clehastheform: Iout S S 0 0 I∗ 3.2.Polarizationformalism,scattering,andphotondetection 11 12 Qout 1 S S 0 0 Q∗ Taonuhdsewtphooelriarkrtidrzaeanstsicofronibrmepdraotepi.oegnr.tidienusrFionifsgcthhsceear,tptHheroeitnnognnisenvgse,anm&tspYlreeodlrykineon(S1p9r9ev4i)-. UVoouutt= k2d2  0012 0022 −SS3334 SS3444UV∗∗. (11) Thetheoreticalbasisfortheformalismpresentedinthissection The entries of the Mueller matrix are obtained by simple canbefoundinBohren&Huffman(1983). relations from the elements of the scattering matrix, S (θ) and 1 If we considera photonbeingscattered off a sphericalpar- S (θ). 2 ticle(seeFig.2),theoutgoingelectromagneticwaveassociated The angle-dependent classical intensity distribution of a with the photon can be resolved into two components, E and scatteredelectromagneticwavemeasurestheprobabilityoffind- k E . These components refer to directions of the electric-field ingascatteredphotonatagivendirection.Suchprobabilityden- ⊥ vector parallel and perpendicular to the scattering plane. For sity distributions are derived from equation (6). An important Goosmann&Gaskell:ModelingAGNPolarization 5 3.3.Computationofdustproperties y z Mathis,Rumpl,&Nordsieck(1977,MRN)suggesteddustcom- Ex* positionstoreproduceextinctioncurvesobservedinourGalaxy. z φ’ x Eyout Exout Tbuhteioynaspsruompoerdtivoanrailoutostaysp,ewsiothf dausbteginrgainthsehagvrainingraadsiizuesdainsdtris- Exin an arbitrarypower-law index.Our parameterizationof the dust 180° − γ Θ Second turn by Θ propertiesfollowsthatofMRNandgivesagooddescriptionof i observedGalacticextinctioncurves.Theusercanchoosethear- y Ey* Ex* bitraryminimumandmaximumradiiofthegrain-sizedistribu- φ’ First turn byφbringingEx tion,itspowerlawindex,andtherelativeabundancesofgraphite into the new scattering plane and“astronomicalsilicate”. x Exin Eyin extinTchteiornescurlotsssfrsoecmtioMnsie, aslcbaetdteorsin,gantdheeolreym,ei.net.s, socfatthteerisncgatatenrd- ing matrix, are computed using the code given by Bohren & Huffman(1983). We importedcomplexdielectric functionsfor graphite and silicate measured by Draine & Lee (1984). For Fig.2. Geometry and denotations for a single scattering event. graphite,twodielectricfunctionshavetobeconsideredsincethe TheinsetshowsthefirstrotationoftheE-vectorbytheangleφ, opticalpropertiesfor lightpolarizedparalleland perpendicular theviewonthepolarizationplaneisalongthenegativez-axis. to the crystals axis differ from each other. The code therefore workswith the two differentgraphitetypes having abundances in a ratio of 1:2. It computes a weighted average for the dust aspectincludedinSisthatthescatteringdirectionissam- compositionand grain size distribution defined. The procedure pled dependingon the incident polarizationvector. The degree isdescribed,forexample,inWolf(2003).Thepropertiesofthe P and the position angle γ of the polarization before scatter- resulting“standarddustgrain”arethenusedbyalldust-related inigarecomputedfromtheiincidentStokesvectorandenterthe routinesofS. samplingequationsofθandφ=φ′+180◦−γ: We confine ourselves to using standard Galactic dust such i as is seen in the solar neighborhood,even though there is evi- dence that the tori of AGNs mighthave differentcompositions θ andgrainsizedistributions(seeCzernyetal.2004,Gaskelletal. P(θ) = N |S1(θ′)|2+|S2(θ′)|2 sin(θ′)dθ′, (12) 2004,andGaskell&Benker2006).FollowingWolf&Henning Z 0 (cid:16) (cid:17) 1999,weparameterizeGalacticdustbyamixtureof62.5%car- P (φ′) = 1 φ′− |S1(θ)|2−|S2(θ)|2P sin2φ′ . (13) bonaceous dust grains and 37.5% “astronomical silicate”. We θ 2π |S (θ)|2+|S (θ)|2 i 2 ! considergrainradii, a, from 0.005µm to 0.250µm with a dis- 1 2 tributionn(a) ∝ as with s = −3.5.The resultingcross-sections The number N is a normalizationconstant in orderto have andthealbedoareshowninFig.3asafunctionofwavelength. (12)rangefrom0to1forscatteringanglesbetween0◦and180◦. Thefigureshowsthatforthisparticulardustmodel,thealbedo Tosampleφandθ,therighthand-sidesoftheseequationshave isratherflatwithavalueof0.55–0.6overthewavelengthrange tobesetequaltorandomnumbersr ,r .Theequationsarethen considered.Forwavelengths.2500Åitfallsto0.4.Thecross- 6 7 solvedfortheangles. sectionsalldecreaseregularlywithwavelength,withtheexcep- Notethatthesamplingisindependentoftheincidentpolar- tionofthewell-knownhumparound2175Å. ization for θ but notfor φ. In severalMonte-Carlopolarization transfer codes describedin the literature, the incidentpolariza- 4. Simulationoftorusgeometries tion does not affect the sampling of the scattering angles. This does not present a problem if one considers unpolarized inci- Inthissectionweinvestigatehowmuchofthepolarizationprop- dent radiation and low optical depths. Also for very high op- ertiesoftype-1andtype-2AGNscanbeproducedbyauniform- tical depths, when multiple-scattering neutralizes the polariza- density torus alone. Kartje (1995) modeledthe polarizationin- tioninsidethescatteringregion,theincidentpolarizationcanbe ducedbyscatteringoffacylindricallyshapedtorus.Theirtorus neglected. However, results for intermediate optical depths are model was adopted from a fit to NGC 1068 given by Pier & sensitive to the sampling method and they should consider the Krolik (1992). This torus is geometrically rather compact and polarizationstateoftheincidentphoton. islocatedwithina radiusof1pcfromthecentralsource.Such Whenaphotonescapesfromthemodelregionitisrecorded a cylindrical torus is not necessarily physical, so we examine byoneofthevirtualdetectors.Itisthennecessarytorotatethe whether the results of Kartje can be confirmed with more gen- polarizationplanearoundtheflightdirectionuntilitmatchesthe eraltori,andweextendtherangeofparameterspaceexplored. referenceaxisofthedetector.TheStokesvectorsofallincoming photonscanfinallybeaddeduptothevaluesIˆ,Qˆ,Uˆ andVˆ.The 4.1.Curvedsurfacesversussharpedges netpolarizationpropertiesarederivedfrom: The dusty tori examined by Kartje (1995), Wolf & Henning (1999), Young (2000), and Watanabe et al. (2003) have rather Qˆ2+Uˆ2+Vˆ2 sharpedges,and,sincewefindthatpolarizationresultscande- P = , (14) p Iˆ pendstronglyongeometricaldetails,wehaveinvestigatedaless artificialtorusgeometrywith an ellipticalcross-section.To ex- 1 Uˆ γ = arctan . (15) aminetheinfluenceofsharpedgesofthecylindricaltorusonthe 2 Qˆ polarization,wedefineatoruswithsimilardimensions,andthe 6 Goosmann&Gaskell:ModelingAGNPolarization 1.0 0.9 0.8 0.2 0.7 o 0.6 d e 0.5 b al 0.4 0.3 P 0.2 0.1 0.1 0.0 6×10-12 2cm]5×10-12 0 on [4×10-12 secti3×10-12 oss-2×10-12 1 cr 1×10-12 0 1100--11 2000 4000 6000 8000 10000 λ [Angström] fFoirg.t3h.isCphaapraecrtearsisaticfupnrcotpioenrtioefsowfatvheelednugstthc.oTmoppo:sailtbioendoadvoapltueed. F / Fcent 10-2 Bottom: cross-sections for extinction (black, solid), scattering 1100--33 (red,dashed),andabsorption(blue,dotted). 10-4 2000 3000 4000 5000 6000 7000 8000 λ [Angstrom] Fig.5. Modeling a cylindrical torus with an elliptical cross- section and θ = 30◦ (see section 4.1). Top: polarization, P. 0 Bottom: the fraction, F/F , of the centralflux, F , seen at dif- ∗ ∗ ferentviewinginclinations,i.Legend:i = 87◦ (edge-on)(black crosses), i = 76◦ (orangetriangles with points down), i = 70◦ (intermediate) (maroon stars), i = 57◦ (purple triangles with points up), i = 41◦ (green diamonds), i = 32◦ (red squares), andi=18◦(face-on)(bluecircles). cosi because it givesequal flux per bin for an isotropic source located at the center of the model space if there is no scatter- ing.Ourfigureisquitesimilartothecorrespondingdiagramsin Fig.4. Geometry of the three torus models we consider: (1) Kartje‘spaper(seehisFig.5). the cylindricaltorus used by Kartje, (2) a compact elliptically- TheonlydifferencebetweenourresultsandthoseofKartje shapedtorus, and (3) an extendedellipticaltorus. All tori have is that we generally obtain slightly lower polarization degrees thesamehalf-openingangleΘ0 andaslightlydifferentwavelength-dependentslopeforthescat- tered flux. This can be explained by the fact that we calculate ourcross-sectionsfromMie theoryof a specific dustcomposi- same optical depth in the V band (τ ) ∼ 750 along the radius tionwhilstKartjeusedcross-sectionsgivenbyMezger,Mathis, V in the equatorial plane. Thus, practically no photon is able to &Panagia(1982). penetratethroughthetorusandonlyscatteringoffitssurfaceis Wealsoinvestigatedthepolarizationofacompacttoruswith relevant.However,ourtorushasan ellipticalcross-section(see an elliptical cross-section for changing θ0. Again we obtained Fig.4)insteadoftherectangularcross-sectionusedbyKartje. similar results (not shown) to those for Kartje‘s cylindrically- Our results compare very well to those obtained by Kartje shaped tori. Thus, the differences in polarization between the (1995). In Fig. 5 we show polarization and flux (normalized ellipticalandcylindricaltoriarenegligible.Havingsharpedges to the flux of the centralsource)versuswavelengthat different in the cylindrical model rather than the more realistic rounded viewingdirections.Thetorusconsideredhasahalf-openingan- edgesoftheellipticaltorusdoesnotintroducespuriouseffects. gleofθ = 30◦.Thepositivevaluesof Pdenotethatthepolar- 0 ization vector is orientedperpendicularlyto the symmetry axis 4.2.Theeffectoftheshapeoftheinneredgeofthetorus (type-2polarization).In our simulationsthe torus is filled with standardGalacticdust,parameterizedasdescribedattheendof A real torus is undoubtedlythicker than the geometricallythin section3.Wesampleatotalof108photonsandrecordspectraat cylindrical torus of Kartje. Direct imaging of NGC 4261 (= 10differentviewinganglesscaledincosi,whereiismeasured 3C 270) shows that the dusty torus in that AGN extends out fromtheaxisofthetorus.We showourresultsasafunctionof to 230 pc (Ferrarese, Ford, & Jaffe 1996). A similar dust lane Goosmann&Gaskell:ModelingAGNPolarization 7 acrossthenucleusofM51(=NGC5194)extendsby∼100pc (Fordetal.1992).Theinnerradiioftoriareobtainedbyinfra- red reverberationmapping of the hot dust and are in the range oftenstohundredsoflight-daysforSeyfertgalaxies(seeGlass 0.2 2004andSuganumaetal.2006). Theouterregionsoftorihaveconsiderableopticaldepth,so theirpreciseshapeisunimportant,sincenophotonsescapepar- alleltotheequatorialplaneofthetorus.Theshapeoftheinner P 0.1 regionfacingthecentralenergysourceismorerelevant.Current torus models commonly consider inner surfaces that are con- vex towards the central source. We thus model optically-thick, uniform-density tori with elliptical cross sections, an inner ra- dius of 0.25 pc, and an outer radius of 100 pc. We compare 0.0 these results to the modelingof a more compacttoruswith the same half-opening angle, θ = 30◦, as in section 4.1. We de- 0 termine the dust density by fixing τ at ∼ 750. Variability ob- 1 V servationsimplythatthe size ofthe opticalandUV-continuum sourceinSeyfertgalaxiesislessthanafewlight-days,asisalso 1100--11 expectedfromsimpleblack-bodyemissivityarguments.Hence, npwiohtieennts-ilcziokeneoseifdmtehirseisnicgoonsncrtaeintgtueiourinmn.gNsoooffutertchteheaittnotrhouiussr,cwmoenoscdiadenlerananetidgolneacsrsteutmhmeaeifinas- F / Fcent 10-2 validforobjectswithhigherluminositiesbecauseboththesize 1100--33 of the centralemission regionand the inner radiusof the torus scalewithluminosity. 10-4 The resulting spectra at different inclinations are shown in Fig.6.Iftheviewingangle,i,islessthanθ0(thuscorresponding 2000 3000 4000 5000 6000 7000 8000 toa type-1object),we onlyobservearegulartype-1spectrum. λ [Angstroem] Wefindthatthereisnosignificantpolarizationinthiscase.Ifwe look at a type-2 objectat a higher inclination angle, only scat- Fig.6.Modelingalargetoruswithanellipticalcross-sectionand tered (and hencepolarized)lightis detected.This is analogous θ = 30◦ (see section 4.2). Top: polarization, P. Bottom: the 0 to the results obtained for the compact torus shown in Fig. 5. fraction,F/F ,ofthecentralflux, F ,seenatdifferentviewing ∗ ∗ Theoverallshapeofthepolarizationspectrumforbothsizesof inclinations, i. Legend: i = 70◦ (intermediate) (maroon stars), thetorusisrathersimilaraswell.Withincreasingviewingangle i = 63◦ (pinktriangleswithpointstotheright),i = 57◦ (purple thelevelofthepolarizationspectrumrises,reachesamaximum, triangleswithpointsup),i=49◦(browntriangleswithpointsto anddecreasesagaintowardsedge-onlinesofsight.Theshapeof the left), i = 41◦ (green diamonds),i = 32◦ (red squares), and the P-spectrumdoesnotchangesignificantlybetweendifferent i=18◦(face-on)(bluecircles) type-2inclinations. Therearedifferencesbetweenourresultsofmodelingalarge torus (case 3 in Fig. 4) with half-openingangle θ = 30◦, and 0 the analogous compact torus (case 2 in Fig. 4) with identical half-openinganglebutsmallerdimensions.Astrikingdifference occurs in the angular flux distribution: the large torus scatters considerably fewer photons towards an observer at intermedi- ateviewinganglesbecausetheyhittheouterpartsofit(seethe illustration in Fig. 7). Towards edge-on viewing directions the probability of seeing scattered photons is much lower than for thesmalltorus.Thespectralslopeofthescatteredradiationalso differsbetweenthetwotori.Whilethespectrumisflatinthecase of a compacttorusit rises towardsthe blue for the large torus. This can be explained by the increasing tendency of forward- Fig.7.Comparisonofthecompactandtheextendedtoruswith scatteringatshorterwavelengths.Photonsescapingathigherin- θ0 =30◦intheV-band. clinationshavetoundergoback-scattering;thisismorelikelyto happenatlongerwavelengths. There are also differences in the polarization signatures of sufficientstatistics.Ontheotherhand,itclearlyfollowsfromour both tori. Although the overallspectral dependenceof P is the computationsthatthespectralfluxatanglesi > 76◦ isreduced same, the level of P is changed. The strongest changes are at by a factor of almost ∼ 2×107 with respect to the flux of the higher inclinations when the central source is becoming ob- source.Therefore,thepolarizedfluxattheseanglesisverylow. scured by the torus. As with the total flux (see above), for the WeshowthedifferencesinV-bandtotalfluxandpolarization largertorus,Pissignificantlylower(comparethecaseofi=70◦ betweenalargeandasmalltorusinFig.8.Thetoppanelshows betweentheupperpanelsoffFig.5andFig.6).Foralargetorus, thepolarizationasafunctionoftheviewingangle,andthebot- our current models sampling several 109 photons do not con- tompanelthefractionofthelightreachingtheobserver.Aswas strainthepolarizationwellatveryhighinclinations.Thenumber shown above,the differencesbetween the two torusshapes are ofphotonsscatteredintothesedirectionsistoosmalltoallowfor mostimportantat higherinclinations.At i ∼ 70◦ the degreeof 8 Goosmann&Gaskell:ModelingAGNPolarization 0,15 0.2 0,1 P 0,05 0.1 P 0 0.0 -0,05 -0,1 1 1100--11 -0,15 2000 3000 4000 5000 6000 7000 8000 F / Fcent 10-2 λ [Angstrom] Fig.9. Polarization averaged over type-2 viewing angles (see 1100--33 section4.3).ApositivevalueofpolarizationdenotesanE-vector orientedperpendiculartothe torussymmetryaxis;fornegative 10-4 valuesthe E-vectoris alignedwith the projectedaxis. Legend: θ = 10◦ (blackdashedline),θ = 20◦ (solidredline),θ = 30◦ 0 0 0 40 45 50 55 60 65 70 (green dot-dashed line), θ = 45◦ (blue dots), θ = 50◦ (long 0 0 yellow dashes), θ = 60◦ (browndouble dots and dashes), and Inclination [deg] 0 θ =75◦(pinkdouble-dashesanddots). 0 Fig.8.Differencesbetweenlargeandsmalltoriwithanelliptical cross-sectionintheV-band(seesection4.2).Top:polarization, P, and bottom, the fraction, F/F , of the central flux, F , as a ∗ ∗ functionofviewinginclinations,i.Thedashedlinesdenotethe Varying the opening angle shows several important things. thin elliptical torus (case 2), the solid line the extended torus Forθ < 53◦ theabsolutevalueofthepolarizationdecreasesas 0 (case3) theopeningangleincreases(seeFig.9),aswasfoundbyKartje (1995)forcompacttori.Thepolarizationvectorisorientedper- pendicularlyto the axis for all viewing directions i > θ , as is 0 observed in type-2 AGN. For θ > 60◦, only parallel polariza- 0 polarizationreachesadifferenceof6%,andthefluxdiffersbya tionvectorscanbeseenatviewinganglesi>θ .Inthisrangeof 0 factorofalmost100. θ theabsolutedegreeofpolarizationincreaseswiththeopening 0 angle. The reasonfor the flip of the relative position angle can be 4.3.Theeffectofthetorushalf-openingangle explained by the scattering phase function, and by the geome- Kartje (1995)has shownthatthe half-openingangle,θ , of the try of the inner parts of the torus (Kartje 1995). For a distant 0 torusisanimportantparameterfortheobscurationandreflection observerlookingatthetorusalonganoff-axislineofsight,the properties.Whilemodelinglargetori,weexaminehalf-opening scattered radiationcomesfrom the inner surface walls. In part, angles ranging from 10◦ to 75◦. Variation of θ is realized by these consist of the inner torus wall facing the observer most 0 changing the vertical half-axis of the elliptical torus cross sec- directly, but they also consist of the two surfaces on the side. tion.Theothermodelparametersaredefinedasfortheprevious Duetothescatteringgeometry,thephotonsscatteredofftheside caseofθ =30◦insection4.2. wallsarepolarizedalongtheprojectedsymmetryaxis,whilstthe 0 photonscomingfromthefarwallareperpendicularlypolarized. The ratio of the solid angle that the far side of the visible in- 4.3.1. Toriwithnarroworwideopenings nersurfacesubtendstothesolidanglethatthevisibleinnerside wallssubtendchangeswiththehalf-openingangleofthetorus, Forlargetori,θ isadominantparameterforboththedegreeof 0 andsodoestheoverallpolarizationvector. polarizationandthepositionangle,γ.InFig.9weshowthepo- larizationofthescatteredradiationasafunctionofwavelength and for variousθ . Due to a similar overallshape of the wave- 0 4.3.2. Transitioncase:intermediatetorushalf-opening lengthdependenceofP,weaveragethepolarizationovertype-2 angles viewing angles, i, with i > θ . We thereby exclude the high- 0 est inclinations with an insufficient number of photons, where Forintermediateopeningangleswith53◦ < θ < 60◦ theorien- 0 the statistics of P are too poor.For viewing angleswith i < θ tation of the polarizationpositionangle seen at type-2viewing 0 (correspondingto type-1 objects seen face-on) the polarization anglesdependsontheexactinclination.Weillustratesuchacase isnegligible. inFig.10,wherewesetθ =57◦. 0 Goosmann&Gaskell:ModelingAGNPolarization 9 0.03 0.15 0.02 0.1 0.01 V-band Peff P 0 0.05 -0.01 -0.02 0 0 20 40 60 80 0 20 40 60 80 Half-opening angle Θ [deg] Half-opening angle Θ [deg] 0 0 Fig.10.Modelinganexpandedtorusatanintermediateopening Fig.11.Effective polarization, P (see section 4.3), fortype-2 eff angle of θ = 57◦. The graphshows the polarizationdegree in viewingpositionsasa functionofthe half-openingangle ofan 0 the visual band versus inclination angle i. A positive value of extendedtorus(case3). polarization denotes an E-vector oriented perpendicular to the torussymmetryaxis;fornegativevaluesthe E-vectorisaligned withtheprojectedaxis. θ outsidethisintervalthewavelength-dependenceofPisrather 0 lowanddoesnotexceedafactorof2.Sincethescatteringcross sectionofinterstellargrainsincreasesstronglyfromtheoptical For a line of sight passing close enough to the horizon of to the UV, wavelength-independent polarization is commonly thetorus(i.e.,wheniisonlymoderatelylargerthanθ0)wefind supposed to be the fingerprintof electron scattering. However, that the polarization vector is parallel, which means that type- scattering in opaque dust clouds produces relatively grey scat- 1polarizationcanbeproducedatobscuredviewinginclinations tering(Kishimoto2001). (Fig.10).Iftheinclinationincreasesfurtherthepolarizationvec- Ourapparentlycontradictoryresultofrelativelywavelength torswitchesbacktotype-2polarization.Itisinterestingtonote independentpolarizationwithdustscatteringarisesbecausewe thatsuchatoruscanproducesignificantpolarizationdegreesup are considering scattering off optically-thick material, and be- to2%forbothorientationsofthe E-vector. cause of the relatively small variation of the albedo over the Inordertoillustratetheintegraleffectoftheopeningangle optical and UV spectral regions (see Fig. 3). The approximate onthepolarization,weplotinFig.11thepolarization,Peff,av- constancyofthealbedoisbecausethescatteringandabsorption eraged overall type-2 viewing positions, and over wavelength, cross-sectionsvaryin asimilar mannerwithwavelength.Since asa functionofthehalf-openingangleofthe torus.Thediffer- we assume an optically-thick torus, we see emergent photons ence between type-1and type-2 polarizationis ignoredin Peff. thathavebeenscatteredatanopticaldepthτ∼1.Thisisregard- TheabsolutevaluesofPareintegrated. lessofwavelength2.Theincreaseinscatteringcross-sectionwith The figure shows that the torus polarizes most effectively decreasing wavelength only means that the shorter wavelength whenhavingeitherasmalloralargehalf-openingangle.Inthe photonswe see havebeenscatteredcloser to thesurfaceof the transitionregionbetweentype-1andtype-2polarization(i.e.for torus. 53◦ < θ < 60◦)theintegratedpolarizationgoesthroughamini- A significant change in albedo with wavelength, however, mum. will cause a color dependencyin the intensity and polarization of the scattered light3. Shortwardsof ∼ 2500Å the albedode- 4.4.Wavelengthinsensitivityofpolarizationduetodust creases,butthisrangeisatthelowerlimitofthespectralrange scattering considered in our modeling. The effect can be seen in the nor- malized flux spectra of the torus models shown in Figs. 5 and Wavelength-independentpolarizationiswidelytakentobeasig- 6. FortheGalacticdustcompositionweimplemented,itisless natureofelectronscattering,butwehaveshownin Figs. 6and visibleinthepolarizationspectra. 9thatdustscatteringcanalsoproducewavelength-independent Another grain property that needs to be considered is the scattering. Thusa flat polarizationcurve is not a uniquesigna- degree of asymmetry of the scattering since this is effectively ture of electron scattering. As Zubko & Laor (2000) pointout, anangle-dependentalbedochange.Towardshorterwavelengths, the wavelengthdependenceof polarizationprovidesa probeof Galactic dust grains are more strongly forward scattering and the grain scattering properties. Inspection of the wavelength- dependentpolarizationcurvesforthelargetorusgeometriescon- 2 Thisisthereasonthatthesunlitsidesofcloudsintheearth‘satmo- sidered above (see Figs. 6 and 9) shows that the polarization sphereareextremelywhite. for half-opening angles with 30◦ < θ0 < 60◦ is wavelength- 3 Thisisthecauseofcolorationsintheatmosphereofthegiantplan- independentovertheopticalandmostoftheUV.Forvaluesof ets. 10 Goosmann&Gaskell:ModelingAGNPolarization the polarization phase function changes (see Draine 2003). As forThomsonscattering,forward-scatteredlighthasalowerpo- 1.0 larizationthansideways-scatteredlight.Thepolarizationspectra obtainedforthetorusmodelsdependonthesephasefunctions. 0.8 Theyadditionallyexplainwhya slightwavelength-dependence of the polarization can be found for very narrow or very wide openinganglesofalargetorus(seeFig.9). 0.6 P 0.4 5. Polarizationfrompolar-scatteringregions Scattering in polar regionsof AGNs has allowed the discovery 0.2 of hidden Seyfert-1 nuclei in type-2 objects by radiation being periscopicallyscatteredaroundtheobscuringtorus.Thecentral 0.0 parts of the polar double cone have to be at least moderately ionizedduetotheintenseradiationfromtheAGN.Themedium 1.0 couldbeassociatedwiththewarmabsorberseeninmanyAGN (see Komossa 1999 for a review). The Doppler shift of the X- 0.8 ray absorption lines indicates that the medium is outflowingat roughly1000km/s.Withincreasingdistancefromthecenter,the 0.6 otdhuuetsfltsuomwbulisvmtebaloeticoointpytriaacdnaidlulyisn,tthdeiunnss,tiatcysootuyflpdteha-el1srooadbbijeaeptcitrosensaerdneetcn.roHetaoosweb.esvBcueerry,eotdhn.ids F / Finc 0.4 5.1.Polarelectronscattering 0.2 Thepolarizationinducedbyscatteringinpolar,conicalelectron- 0.0 scatteringregionshasbeenthesubjectofseveralpreviousstud- 10 20 30 40 50 60 70 80 90 ies.Brown&McLean(1977)developedaformalismtocompute Inclination i [deg] the polarization expected from scattering inside optically thin, Fig.12. Modeling double polar cones of various half-opening axisymmetricscatteringregions.Thisformalismwasappliedby angles.Top:polarization,P,withpositivevaluesdenotingtype- Miller&Goodrich(1990)andMilleretal.(1991)tocomputethe 2polarization(perpendiculartothesymmetryaxis).Bottom:the polarizationforpolarscatteringcones.Wolf&Henning(1999) fraction,F/F ,ofthecentralflux.Bothvaluesareplottedversus ∗ andWatanabeatal.(2003)extendedthemodelingtooptically- theinclinationiwithrespecttotheobserver.Thedifferentsym- thick material using Monte-Carlo techniques that can account bols denote different half-opening angles of the double-cone. formultiplescatterings. Legend:θ = 10◦ (black circles), θ = 30◦ (red squares), and WeconfirmsuchresultsinFig.12usingS.Thefigure θ = 45◦ (Cbluediamonds).TheopticCaldepthbetweentheinner C showsthedegreeofpolarizationandthetotalfluxasafunction andoutershelloftheconesissettoτ =1. es of the observer‘s inclination for an electron scattering double- coneofuniformdensityandwiththeopticaldepthτ =1.This es optical depth is measured in the vertical direction between the law.ComparisonofFig.13withFig.7ofWatanabeetal.(2003, innerandtheoutershellofonecone.Inordertoisolatetheef- bottompanel)showsthatthedifferencein Pisverysmallwith fectsofthescatteringconefromthepolarizationinducedbythe respecttoauniform-densitytorus. disk,weuseananisotropicallyemittingcentralsourcewiththe emission angles being restricted to the solid angle defined by the scattering cones. The three curves denote the half-opening 5.2.Polardustscattering angles θ = 10◦, θ = 30◦, and θ = 45◦. The inclination is C C C measuredfromthesymmetryaxisofthedouble-cone. Beyond the dust sublimation radius the scattering cone could As expected, polar electron-scattering cones produce type- containdust.Weinvestigatedthisusingasimilarbi-conicalge- 2 polarization directed perpendicularlyto their symmetry axis. ometry and our Galactic dust prescription – see section 3. In Thedegreeofpolarizationrises fromface-onto edge-onview- Fig. 14 we show the polarization and total flux resulting spec- ingangles.Thelattereffectisduetotheangle-dependentpolar- tra for a centrally-illuminateddust cone seen at differentview- ization phase function of Thomson scattering. For wider open- ing angles i. The half-opening angle of the cone has been set inganglesofthecones,thenetpolarizationPdecreasesbecause toθ = 30◦,anditsopticaldepthintheV-bandtothemoderate C it is the result of integrating a broaderdistribution of polariza- valueofτ =0.3.Thestrongwavelengthdependenceofthedust V tion vectors. The breaks of the polarization curves at i = θ extinctionproperties(seeFig.3)isclearlyvisibleinthefigure. C in Fig. 12 are due to the impact of multi-scattering inside the It differs for polar viewing angles, which cross the cone, from cones, the analogous breaks in total flux curves are due to the thosealongequatorialdirections.Theformeronesshowthedust angle-restrictedcentralemission. extinctionseenintransmission,whilethelatteronesshowdust InFig.13weplottheinfluenceoftheopticaldepthonthepo- reflection. larizationforthepolar-coneswithθ = 30◦.Thevariouscurves ThetotalfluxshowninthebottompanelofFig.14issignifi- C denote different optical depths. A similar case was considered cantlyreddenedwheni<θ .Inadditiontothat,thewellknown C by Watanabe et al. (2003). The density of their electron cones extinctionfeatureat2175µmisseen.Itsdepthdecreaseswithin- varies with the distance from the center according to a power creasinginclination.Inreflection(i.e.,alongequatorialviewing